Two beautiful theorems about C(X)

Let be a topological space. Let be the set of continuous functions from to . can also be thought of as set of smooth sections of the trivial bundle . Anyway, we get a contravariant functor

where is the category of . is the category of with and every sends to . The two beautiful to be discussed here are the following. (Hewitt)For , compact Hausdorff there is a bijection

(Swan)If is compact Hausdorff then taking sections gives a bijection from

These two beautiful theoremshave some remarkable consequences If , are compact and Hausdorff then,

\textup{2} leads to the following result in –

is a consequence of the following Let be compact, Hausdorff topological space

For ,

is a maximal ideal,

If is a maximal ideal, then such that ,

where is the set of all maximal ideals of a ring equipped with topology. The isomorphism takes to

Proof:

Clearly is maximal as which is a field.

Notice, if , then If is an such that for all in then for every , such that . Each there exists such that . Since is compact cover . Using which do not vanish on respectively, define

. Observe, . Define . Clearly and . Thus . Thus the only maximal ideals of is of the form for some .

For any ideal of a ring define,

is the basis for all sets in the space under topology. The map

which sends

is already a bijection. All we need to show is

IF closed then define and \vspace{5pt}
IF be a basic closed set in , ie, for some then, define

. Then is clearly a closed set and clearly .

Proof:(of Theorem 1) In fact the gives the map between the sets of the respective category.One-one
Let . Then

, where . If then

by using bump functions near each pointOnto
Given a map , we induce a map

By \textup{5} we get a map

It is clear that . Proof:( sketch of proof of theorem 2)
Notice that Let be the map
G: isomorphism class of vector bundles over finitely generated C(X)-modules
where given a vector bundle = smooth sections of .
Since is compact, any vector bundle is a of a trivial bundle of finite dimension, ie . Hence is a sub-module of due to the following isomorphism. smooth sections on the trivial bundle
Thus is a finitely generated module. Moreover every bundle of finite dimension over a compact space has a complement, say , hence . Hence its projective. Given a finitely generated projective module over , say , find and a module , such that